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# What is the range of the roots of ||x – 1| – 2| = 1?

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Math Expert
Joined: 02 Sep 2009
Posts: 59721
What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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29 Aug 2017, 02:02
3
21
00:00

Difficulty:

55% (hard)

Question Stats:

62% (01:50) correct 38% (01:42) wrong based on 454 sessions

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Fresh GMAT Club Tests' Challenge Question:

What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

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Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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30 Aug 2017, 03:52
10
Bunuel wrote:
What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

Another approach

Let $$z = x -1$$

$$||z| – 2| = 1$$.............square both sides

$$z^2-4|z|+4=1$$

$$z^2-4|z|+3=0$$

$$(|z|-3)(|z|-1)=0$$

$$|z|=3$$ or $$|z|=1$$

substitute z from above

$$x–1=3$$ or $$x–1=-3$$
$$x=4$$ or $$x=-2$$

OR

$$x–1=1$$ or $$x–1=-1$$
$$x=2$$ or $$x=0$$

Range = Max value - min value

$$R=4-(-2)=4+2=6$$

##### General Discussion
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Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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29 Aug 2017, 02:44
4
3
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

||x – 1| – 2| = 1
|x – 1| – 2 = 1 or |x – 1| – 2 = -1

if |x – 1| – 2 = 1
|x – 1| = 3
x – 1 = 3 or x – 1 = -3
giving us x = 4, -2

if |x – 1| – 2 = -1
|x – 1| = 1
x – 1 = 1 or x – 1 = -1
giving us x = 2, 0

now we have x=-2,0,2,4
range will be 4+2 = 6
D
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What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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31 Aug 2017, 20:22
1
1
The expression is in the form ||a| - 2| = 1

There are 4 possibilities, when trying to solve this equation
|-a -2| = 1
There are 2 options:
-(-a -2) = 1 => a +2 = 1 => a = -1
(-a - 2) = 1 => -a = 3 => a = -3

|a - 2| = 1
There are 2 options:
-(a - 2) = 1 => -a + 2 = 1 => a = 1
(a - 2) = 1 => a = 3

Hence, there are 4 values for a which are -3,-1,1,3

Coming back to the question, if we substitute a to be x-1

x - 1 = -3 => x = -2
x - 1 = -1 => x = 0
x - 1 = 1 => x = 2
x - 1 = 3 => x = 4

Hence, the range of the values is 6(Option D)
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Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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31 Aug 2017, 22:20
1
the values are
X-3=1
-X+3 = 1
-X-1 = 1
X+1 = 1

the values X are { 4, 2, 0, -2}

hence range = 4-(-2)
6
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Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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02 Sep 2017, 00:34
2
||x-1|-2|=1

Removing the outer modulus, we get
|x-1|-2 = 1 (or) -1

|x-1|= 3 (or) 1

Now, if we remove the modulus for |x-1| we will get the following four possible values for x

x-1= 3 (or) -3 => x= 4 (or) -2
X-1= 1 (or) -1 => x= 2 (or) 0

Hence the four possible values of x are (-2,0,2,4)
=> Range= 4- (-2) = 6

Ans-> Option D
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What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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17 Sep 2018, 03:54
1
I just applied a rule for complex absolute value equations:

If the equation contains 1 variable and 1 or more than 1 constant(s) in 1 or more than 1 absolute value expressions, the only two cases we have to consider are (1) that the expressions have the same sign and (2) that they have different signs.

Applied to the question here I computed the following:

case (1):

(x-1)-2=1
x-3=1
x=4

case (2)
-(x-1)-2=1
-x+1-2=1
-x-1=1
-2=x

Since we are asked for the range (= highest value-lowest value), we have to perform this final step to get to the answer: 4-(-2)=6.

Please hit Kudos if you liked this approach
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Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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24 Dec 2018, 04:53
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

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What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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Updated on: 08 Oct 2019, 04:00
1
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

Case 1: |x-1|-2 = 1
|x-1| = 3
x = 4 or -2

Case 2: |x-1|-2 = -1
|x-1| = 1
x = 2 or 0

Set of roots x = {-2,0,2,4}
Range of roots = 4 - (-2) = 6

IMO D

Originally posted by Kinshook on 08 Oct 2019, 03:13.
Last edited by Kinshook on 08 Oct 2019, 04:00, edited 1 time in total.
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What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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08 Oct 2019, 03:31
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

lx-1l=y
ly-2l=1
y-2=1 or y-2=-1
y=3 or y=1
lx-1l=3 or lx-1l=1
x-1=3 or x-1=-3 or x-1=1 or x-1=-1
x=4 or x=-2 or x=2 or x=0
x=-2 smallest
x=4 largest
range=4-(-2)
D:)
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What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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08 Oct 2019, 03:56
Slope of each line is 1 or -1

Hence the range of the roots of $$||x – 1| – 2| = 1$$= 1+1+1+1+1+1=6 (clearly seen from the graph)

Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

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Posts: 968
Location: United States
Re: What is the range of the roots of ||x – 1| – 2| = 1?  [#permalink]

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11 Nov 2019, 08:51
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

What is the range of the roots of $$||x – 1| – 2| = 1$$?

A. 0
B. 2
C. 4
D. 6
E. 8

|x-1|≥0 when x≥1 (positive); |(x-1)-2|=1…|x-3|=1;
|x-3|≥0 when x≥3: (x-3)=1…x=4=valid (x≥3)
|x-3|<0 when x<3: -(x-3)=1…-x+3=1…x=2=valid (x<3)

|x-1|<0 when x≤1 (negative); |-(x-1)-2|=1…|-x-1|=1
|-x-1|≥0 when x≤-1 (positive): (-x-1)=1…-x=2…x=-2=valid (x≤-1)
|-x-1|<0 when x>-1 (negative): -(-x-1)=1…x+1=1…x=0=valid (x>-1)

valid solutions: x={4,2,-2,0} range=largest-smallest=4-(-2)=6

Ans (D)
Re: What is the range of the roots of ||x – 1| – 2| = 1?   [#permalink] 11 Nov 2019, 08:51
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